Randomized Linear Algebra Approaches to Estimate the Von Neumann Entropy of Density Matrices

The von Neumann entropy, named after John von Neumann, is the extension of classical entropy concepts to the field of quantum mechanics and, from a numerical perspective, can be computed simply by computing all the eigenvalues of a density matrix, an operation that could be prohibitively expensive for large-scale density matrices. We present and analyze two randomized algorithms to approximate the von Neumann entropy of density matrices: our algorithms leverage recent developments in the Randomized Numerical Linear Algebra (RandNLA) literature, such as randomized trace estimators, provable bounds for the power method, and the use of Taylor series and Chebyschev polynomials to approximate matrix functions. Both algorithms come with provable accuracy guarantees and our experimental evaluations support our theoretical findings showing considerable speedup with small accuracy loss.